Question

Question 6 (total 16 points). Sam and Erica are starting a new restaurant in Portland, Oregon. While Sam plans to do the cooking himself, he will need to employ workers and machinery to produce food. He estimates his production function as: q = 1520.25 K. Sam is able to accumulate $10,000 to finance the business. Workers cost $10 per unit and capital costs $50 per unit a. (11 points). If Sam wishes to produce the most output with the finances available, how much labor and capital should Sam employ? Use a Lagrangian to solve this problem. b. (5 points). Does this bundle of capital and labor also minimize the costs? Question 7 (total 15 points) A firm has a production function, q = 20.25 0.75. It wants to minimize cost for a given production q. The wage rate and rental rate on capital are w and r, respectively. a. (4 points). Write the Lagrangian expression for the cost minimization problem b. (5 points). Write the first-order conditions c. (6 points). Find the cost minimizing L and K.

Answer 1

Question 6 - Production function est q = 15L 0.25k here to tal cost (Te)= 10,000 workers cost or wage (W) = $10 / mit and capital costs (2) = $50 fuit. Huse the to the cost for clion ist TC= WL tak => 10,000 = 10L +50k a Now, out put manimisation Lagrangian problem can be > writhe ast La 1540.25% + 1 [10,000-104-5067 Now, the first or du condition will be! [here & is the lagrange multipliu]. OLE 15 (0.25) L*°** lera 5,-0.75 k = 101 - ---0) and at = 15 20:25 - 502=0 => 1520-2=X -- Now, equating both valus of x from equation is and ii) weget? 50 To => - 3.75 -0.75 3.75 2-0.75K. 10 SO 1570.25 3.75L-0.75 k 1540.25 3.75 -0.75% = (-0.75 ☺ - 3.75 K = 320.25 3.75 k = 3 2 2 1.25K - L --- lii) -31 Scanned with CamScanner

WOW ) ahs 10,000 -10L- 50k =0 Now substituting L=1.25k we get? => 10,000 - 1061.25k) -50 K = 0 => 10,00D - 62.5K = 0 -7 K = 160 And as L=1.25k = 1.2 Auceca Sam employs 200 labor and 160 200 nits of capital Camscanner

6 Now condition for cost minimisation is MPL = 1 2L Hue production function q = 15 20 25k marginal product of labor (MPL) = aa = 15 (0,25) 2 elus 70.75 = 3.75Ľ-0.75k ar MPK in to tal cost function and marginal product of capital (MPK) zda 15 10:25 Now, MP2 = " imblis. - 3.752-0175k 15 L0.25 -> 0:25 K - Ś => 1,25KEL Now substituting L=12 L=1.25k wcques 10,000 = 10L+SOK :) 10,00 = 10(1.25k) +50k -> k = 160 . And as L=1.ask = 1.25X 160- Huce this bundle ire. L= 200 k=160 also can minimize the cost. = 200 CamScanner

Question? - production for iton es q = 10.25 ko.75 here, wage rate= W and rental rate on capital: a Lagrangian for cost minimization problem is - l = (WLtok) tar q- 20.25k 0.75 There to tal cost TC = WLtok] e first order Condition will be! 0.x2-0.75 K0.75= -w- al =) W 0.252-0.75k0.75= 1 anin 60 and oth = -0.75 720.25k-0.25_o O - n - 0.75 20 25k-0.25 = x-alig equating vales of a from both equation (i) and (ii) we un o 0.25 1-0.75 0.75 = 0.7520.25k-dias 0.75 WL 0.25k-0,25 -0.25 & 2-0.75 0.75 1. 3t 3 w Lo. 25 k-0.25 = 84-0.75075 3LW = gok 34W = k -- liii) at = q- 20,25 K0175= q- 10.25 * 0.75 Now Scanned with CamScanner

Now substituting K= 3W we get? q = 10:25 (345) 07 => q = 2.28 219 220:* => quts = congo en qe ) 0:75 = 2.282 -) La 0.4 96) 0.75 and as, K= 3 LW hence K= 30 0.49 10 10.75 =) ke 1:29 () -0.025 * 2 s cost minimizing L and k will be L = 0.4 q (2 0.75 and K = 1.2 q ( ) Scanned with CamScanner